Ant Colony Optimization

1. Ant Colony Optimization Optimisation Methods

2. Overview

3. ACO Concept • Ants (blind) navigate from nest to food source • Shortest path is discovered via pheromone trails • each ant moves at random • pheromone is deposited on path • ants detect lead ant’s path, inclined to follow • more pheromone on path increases probability of path being followed

4. ACO System • Virtual “trail” accumulated on path segments • Starting node selected at random • Path selected at random • based on amount of “trail” present on possible paths from starting node • higher probability for paths with more “trail” • Ant reaches next node, selects next path • Continues until reaches starting node • Finished “tour” is a solution

5. ACO System, cont. • A completed tour is analyzed for optimality • “Trail” amount adjusted to favor better solutions • better solutions receive more trail • worse solutions receive less trail • higher probability of ant selecting path that is part of a better-performing tour • New cycle is performed • Repeated until most ants select the same tour on every cycle (convergence to solution)

6. ACO System, cont. • Often applied to TSP (Travelling Salesman Problem): shortest path between n nodes • Algorithm in Pseudocode: • Initialize Trail • Do While (Stopping Criteria Not Satisfied) – Cycle Loop • Do Until (Each Ant Completes a Tour) – Tour Loop • Local Trail Update • End Do • Analyze Tours • Global Trail Update • End Do

7. Background • Discrete optimization problems difficult to solve • “Soft computing techniques” developed in past ten years: • Genetic algorithms (GAs) • based on natural selection and genetics • Ant Colony Optimization (ACO) • modeling ant colony behavior

8. Background, cont. • Developed by Marco Dorigo (Milan, Italy), and others in early 1990s • Some common applications: • Quadratic assignment problems • Scheduling problems • Dynamic routing problems in networks • Theoretical analysis difficult • algorithm is based on a series of random decisions (by artificial ants) • probability of decisions changes on each iteration

9. Implementation

10. Ant Algorithms

11. Ant Algorithms

12. Implementation • Can be used for both Static and Dynamic Combinatorial optimization problems • Convergence is guaranteed, although the speed is unknown • Value • Solution

13. The Algorithm • Ant Colony Algorithms are typically use to solve minimum cost problems. • We may usually have N nodes and A undirected arcs • There are two working modes for the ants: either forwards or backwards. • Pheromones are only deposited in backward mode.

14. The Algorithm • The ants memory allows them to retrace the path it has followed while searching for the destination node • Before moving backward on their memorized path, they eliminate any loops from it. While moving backwards, the ants leave pheromones on the arcs they traversed.

15. The Algorithm • The ants evaluate the cost of the paths they have traversed. • The shorter paths will receive a greater deposit of pheromones. An evaporation rule will be tied with the pheromones, which will reduce the chance for poor quality solutions.

16. The Algorithm • At the beginning of the search process, a constant amount of pheromone is assigned to all arcs. When located at a node i an ant k uses the pheromone trail to compute the probability of choosing j as the next node: • where is the neighborhood of ant k when in node i.

17. The Algorithm • When the arc (i,j) is traversed , the pheromone value changes as follows: • By using this rule, the probability increases that forthcoming ants will use this arc.

18. The Algorithm • After each ant k has moved to the next node, the pheromones evaporate by the following equation to all the arcs: • where is a parameter. An iteration is a completer cycle involving ants’ movement, pheromone evaporation, and pheromone deposit.

19. Steps for Solving a Problem by ACO • Represent the problem in the form of sets of components and transitions, or by a set of weighted graphs, on which ants can build solutions • Define the meaning of the pheromone trails • Define the heuristic preference for the ant while constructing a solution • If possible implement a efficient local search algorithm for the problem to be solved. • Choose a specific ACO algorithm and apply to problem being solved • Tune the parameter of the ACO algorithm.

20. 1 5 2 4 3 Applications Efficiently Solves NP hard Problems • Routing • TSP (Traveling Salesman Problem) • Vehicle Routing • Sequential Ordering • Assignment • QAP (Quadratic Assignment Problem) • Graph Coloring • Generalized Assignment • Frequency Assignment • University Course Time Scheduling

21. Applications • Scheduling • Job Shop • Open Shop • Flow Shop • Total tardiness (weighted/non-weighted) • Project Scheduling • Group Shop • Subset • Multi-Knapsack • Max Independent Set • Redundancy Allocation • Set Covering • Weight Constrained Graph Tree partition • Arc-weighted L cardinality tree • Maximum Clique

22. Applications • Other • Shortest Common Sequence • Constraint Satisfaction • 2D-HP protein folding • Bin Packing • Machine Learning • Classification Rules • Bayesian networks • Fuzzy systems • Network Routing • Connection oriented network routing • Connection network routing • Optical network routing

23. Advantages and Disadvantages

24. Advantages and Disadvantages • For TSPs (Traveling Salesman Problem), relatively efficient • for a small number of nodes, TSPs can be solved by exhaustive search • for a large number of nodes, TSPs are very computationally difficult to solve (NP-hard) – exponential time to convergence • Performs better against other global optimization techniques for TSP (neural net, genetic algorithms, simulated annealing) • Compared to GAs (Genetic Algorithms): • retains memory of entire colony instead of previous generation only • less affected by poor initial solutions (due to combination of random path selection and colony memory)

25. Advantages and Disadvantages, cont. • Can be used in dynamic applications (adapts to changes such as new distances, etc.) • Has been applied to a wide variety of applications • As with GAs, good choice for constrained discrete problems (not a gradient-based algorithm)

26. Advantages and Disadvantages, cont. • Theoretical analysis is difficult: • Due to sequences of random decisions (not independent) • Probability distribution changes by iteration • Research is experimental rather than theoretical • Convergence is guaranteed, but time to convergence uncertain

27. Advantages and Disadvantages, cont. • Tradeoffs in evaluating convergence: • In NP-hard problems, need high-quality solutions quickly – focus is on quality of solutions • In dynamic network routing problems, need solutions for changing conditions – focus is on effective evaluation of alternative paths • Coding is somewhat complicated, not straightforward • Pheromone “trail” additions/deletions, global updates and local updates • Large number of different ACO algorithms to exploit different problem characteristics